Abstract
Let p(z)=a 0+a 1 z+a 2 z 2+a 3 z 3+⋯+a n z n be a polynomial of degree n, where the coefficients a k may be complex. Then it is obviously of interest to study problems concerning the location of the zeros of the polynomial p(z). These problems, besides being of theoretical interest, have important applications in many areas, such as signal processing, communication theory, and control theory, and for this reason there is always a need for better and better results. In this paper we make a systematic study of these problems by presenting some results starting from the results of Gauss and Cauchy, who we believe were the earliest contributors in this subject, to some of the most recent ones. Our paper is expository.
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References
Affane-Aji, C., Agarwal, N., Govil, N.K.: Location of zeros of polynomials. Math. Comput. Model. 50, 306–313 (2009)
Affane-Aji, C., Biaz, S., Govil, N.K.: On annuli containing all the zeros of a polynomial. Math. Comput. Model. 52, 1532–1537 (2010)
Berwald, L.: Elementare Sätze uber die Abgrenzung der Wurzeln einer algebraischen Gleichung. Acta Sci. Math. Litt. Sci. Szeged 6, 209–221 (1934)
Birkhoff, G.D.: An elementary double inequality for the roots of an algebraic equation having greatest value. Bull. Am. Math. Soc. 21, 494–495 (1914)
Carmichael, R.D., Mason, T.E.: Note on the roots of algebraic equations. Bull. Am. Math. Soc. 21, 14–22 (1914)
Cauchy, A.L.: Excercises de Mathematiques. IV Année de Bure Frères, Paris (1829)
Cohn, A.: Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise. Math. Z. 14, 110–148 (1922)
Datt, B., Govil, N.K.: On the location of zeros of polynomial. J. Approx. Theory 24, 78–82 (1978)
Dewan, K.K.: On the location of zeros of polynomials. Math. Stud. 50, 170–175 (1982)
Diaz-Barrero, J.L.: Note on bounds of the zeros. Mo. J. Math. Sci. 14, 88–91 (2002)
Diaz-Barrero, J.L.: An annulus for the zeros of polynomials. J. Math. Anal. Appl. 273, 349–352 (2002)
Dieudonné, J.: La théorie analytique des polynômes d’une variable. Mémor. Sci. Math. 93 (1938)
Fujiwara, M.: A Ueber die Wurzeln der algebraischen Gleichungen. Tôhoku Math. J. 8, 78–85 (1915)
Gauss, K.F.: Beiträge zur Theorie der algebraischen Gleichungen. Abh. Ges. Wiss. Göttingen, vol. 4, Ges. Werke, vol. 3, pp. 73–102 (1850)
Goodman, A.W., Schoenberg, I.J.: A proof of Grace’s theorem by induction. Honam Math. J. 9, 1–6 (1987)
Grace, J.H.: The zeros of a polynomial. Proc. Camb. Philos. Soc. 11, 352–357 (1901)
Jain, V.K.: On Cauchy’s bound for zeros of a polynomial. Turk. J. Math. 30, 95–100 (2006)
Joyal, A., Labelle, G., Rahman, Q.I.: On the location of zeros of polynomials. Can. Math. Bull. 10, 53–63 (1967)
Kelleher, S.B.: Des limites des Zéros d’une polynome. J. Math. Pures Appl. 2, 169–171 (1916)
Kuniyeda, M.: Notes on the roots of algebraic equation. Tôhoku Math. J. 9, 167–173 (1916)
Kim, S.-H.: On the moduli of the zerros of a polynomial. Am. Math. Mon. 112, 924–925 (2005)
Marden, M.: Geometry of Polynomials. Am. Math. Soc. Math. Surveys, vol. 3. Am. Math. Soc., Providence (1966)
Marden, M.: The zeros of certain composite polynomials. Bull. Am. Math. Soc. 49, 93–100 (1943)
Markovitch, D.: On the composite polynomials. Bull. Soc. Math. Phys. Serbie 3(3–4), 11–14 (1951)
Milovanovic, G.V., Mitrinovic, D.S., Rassias, Th.M.: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)
Montel, P.: Sur la limite supérieure des modules des zéros des polynômes. C. R. Acad. Sci. Paris 193, 974–976 (1931)
Peretz, R., Rassias, Th.M.: Some remarks on theorems of M. Marden concerning the zeros of certain composite polynomials. Complex Var. 18, 85–89 (1992)
Rubinstein, Z.: Some results in the location of the zeros of linear combinations of polynomials. Trans. Am. Math. Soc. 116, 1–8 (1965)
Schur, J.: Zwei sätze über algebraische Gleichungen mit lauter rellen Wurzeln. J. Reine Angew. Math. 144, 75–88 (1914)
Sun, Y.J., Hsieh, J.G.: A note on circular bound of polynomial zeros. IEEE Trans. Circuits Syst. I 43, 476–478 (1996)
Szegö, G.: Bemerkungen zu einem Satz von J.H. Grace über die Wurzeln algebraischer Gleichungen. Math. Z. 13, 28–55 (1922)
Tôya, T.: Some remarks on Montel’s paper concerning upper limits of absolute values of roots of algebraic equations. Sci. Rep. Tokyo Bunrika Daigaku A1, 275–282 (1933)
Walsh, J.L.: An inequality for the roots of an algebraic equation. Ann. Math. 25, 285–286 (1924)
Williams, K.P.: Note concerning the roots of an equation. Bull. Am. Math. Soc. 28, 394–396 (1922)
Zeheb, F.: On the largest modulus of polynomial zeros. IEEE Trans. Circuits Syst. I 38, 333–337 (1991)
Z̃ilović, M.S., Roytman, L.M., Combettes, P.L., Swamy, M.N.S.: A bound for the zeros of polynomials. IEEE Trans. Circuits Syst. I 39, 476–478 (1992)
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Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.
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Affane-Aji, C., Govil, N.K. (2012). On the Regions Containing All the Zeros of a Polynomial. In: Pardalos, P., Georgiev, P., Srivastava, H. (eds) Nonlinear Analysis. Springer Optimization and Its Applications, vol 68. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3498-6_3
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